Optimal. Leaf size=99 \[ -\frac {a^8}{2 x^2}+8 a^7 b \log (x)+14 a^6 b^2 x^2+14 a^5 b^3 x^4+\frac {35}{3} a^4 b^4 x^6+7 a^3 b^5 x^8+\frac {14}{5} a^2 b^6 x^{10}+\frac {2}{3} a b^7 x^{12}+\frac {b^8 x^{14}}{14} \]
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Rubi [A] time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {14}{5} a^2 b^6 x^{10}+7 a^3 b^5 x^8+\frac {35}{3} a^4 b^4 x^6+14 a^5 b^3 x^4+14 a^6 b^2 x^2+8 a^7 b \log (x)-\frac {a^8}{2 x^2}+\frac {2}{3} a b^7 x^{12}+\frac {b^8 x^{14}}{14} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (28 a^6 b^2+\frac {a^8}{x^2}+\frac {8 a^7 b}{x}+56 a^5 b^3 x+70 a^4 b^4 x^2+56 a^3 b^5 x^3+28 a^2 b^6 x^4+8 a b^7 x^5+b^8 x^6\right ) \, dx,x,x^2\right )\\ &=-\frac {a^8}{2 x^2}+14 a^6 b^2 x^2+14 a^5 b^3 x^4+\frac {35}{3} a^4 b^4 x^6+7 a^3 b^5 x^8+\frac {14}{5} a^2 b^6 x^{10}+\frac {2}{3} a b^7 x^{12}+\frac {b^8 x^{14}}{14}+8 a^7 b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 99, normalized size = 1.00 \begin {gather*} -\frac {a^8}{2 x^2}+8 a^7 b \log (x)+14 a^6 b^2 x^2+14 a^5 b^3 x^4+\frac {35}{3} a^4 b^4 x^6+7 a^3 b^5 x^8+\frac {14}{5} a^2 b^6 x^{10}+\frac {2}{3} a b^7 x^{12}+\frac {b^8 x^{14}}{14} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^8}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.17, size = 94, normalized size = 0.95 \begin {gather*} \frac {15 \, b^{8} x^{16} + 140 \, a b^{7} x^{14} + 588 \, a^{2} b^{6} x^{12} + 1470 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 2940 \, a^{5} b^{3} x^{6} + 2940 \, a^{6} b^{2} x^{4} + 1680 \, a^{7} b x^{2} \log \relax (x) - 105 \, a^{8}}{210 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 101, normalized size = 1.02 \begin {gather*} \frac {1}{14} \, b^{8} x^{14} + \frac {2}{3} \, a b^{7} x^{12} + \frac {14}{5} \, a^{2} b^{6} x^{10} + 7 \, a^{3} b^{5} x^{8} + \frac {35}{3} \, a^{4} b^{4} x^{6} + 14 \, a^{5} b^{3} x^{4} + 14 \, a^{6} b^{2} x^{2} + 4 \, a^{7} b \log \left (x^{2}\right ) - \frac {8 \, a^{7} b x^{2} + a^{8}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.91 \begin {gather*} \frac {b^{8} x^{14}}{14}+\frac {2 a \,b^{7} x^{12}}{3}+\frac {14 a^{2} b^{6} x^{10}}{5}+7 a^{3} b^{5} x^{8}+\frac {35 a^{4} b^{4} x^{6}}{3}+14 a^{5} b^{3} x^{4}+14 a^{6} b^{2} x^{2}+8 a^{7} b \ln \relax (x )-\frac {a^{8}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 91, normalized size = 0.92 \begin {gather*} \frac {1}{14} \, b^{8} x^{14} + \frac {2}{3} \, a b^{7} x^{12} + \frac {14}{5} \, a^{2} b^{6} x^{10} + 7 \, a^{3} b^{5} x^{8} + \frac {35}{3} \, a^{4} b^{4} x^{6} + 14 \, a^{5} b^{3} x^{4} + 14 \, a^{6} b^{2} x^{2} + 4 \, a^{7} b \log \left (x^{2}\right ) - \frac {a^{8}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 89, normalized size = 0.90 \begin {gather*} \frac {b^8\,x^{14}}{14}-\frac {a^8}{2\,x^2}+\frac {2\,a\,b^7\,x^{12}}{3}+8\,a^7\,b\,\ln \relax (x)+14\,a^6\,b^2\,x^2+14\,a^5\,b^3\,x^4+\frac {35\,a^4\,b^4\,x^6}{3}+7\,a^3\,b^5\,x^8+\frac {14\,a^2\,b^6\,x^{10}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 100, normalized size = 1.01 \begin {gather*} - \frac {a^{8}}{2 x^{2}} + 8 a^{7} b \log {\relax (x )} + 14 a^{6} b^{2} x^{2} + 14 a^{5} b^{3} x^{4} + \frac {35 a^{4} b^{4} x^{6}}{3} + 7 a^{3} b^{5} x^{8} + \frac {14 a^{2} b^{6} x^{10}}{5} + \frac {2 a b^{7} x^{12}}{3} + \frac {b^{8} x^{14}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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